The relevant fact is that the sequence of free sheaves of \(\mathcal {O}_X\) -modules \(({\mathcal {R}}({\mathcal {F}})^{\cdot }, D^\cdot ) \rightarrow \mathcal {F}\)  is a resolution of \(\mathcal {F}\)  [1]}. This construction can be done for every sheaf \(\mathcal {F}\)  of \(\mathcal {O}_X\) -modules on a topological space \(X\) ; Illusie obtains it as an example of the even more general construction of the standard simplicial resolution of a pair of adjont functors [1]}.

